Problem: A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points \[(-\tfrac32, 1), \; (0,0), \;(0,2),\; (3,0),\; (3,2).\]is an ellipse whose axes are parallel to the coordinate axes. Find the length of its minor axis.
Solution: The four points $(0,0),$ $(0,2),$ $(3,0),$ and $(3,2)$ form a rectangle, and the horizontal line through $(-\tfrac32, 1)$ bisects the rectangle. So, visually, we hope that the center of the ellipse coincides with the center of the rectangle, which has coordinates $\left(\tfrac32, 1\right),$ and that its major axis should pass through the point $(-\tfrac32, 1).$

In this case, the semimajor axis has length $\tfrac32 - (-\tfrac32) = 3.$ Then, its equation must take the form \[\frac{(x-\tfrac32)^2}{3^2} + \frac{(y-1)^2}{b^2} = 1\]where $b$ is the length of the semiminor axis. Since $(0,0)$ lies on the ellipse, setting $x=y=0,$ we have \[\frac{\left(\frac32\right)^2}{3^2} + \frac{1}{b^2} = 1,\]or $\frac{1}{4} + \frac{1}{b^2} = 1.$ Solving for $b$ gives $b = \frac{2\sqrt3}{3},$ so the length of the minor axis is $2b = \boxed{\frac{4\sqrt3}{3}}.$